Puzzle 1
The concept of "defeat", in any case, is not necessarily silly or inapplicable to a particular (game-based) understanding of reasoning, which has always been known to be discursive, so I do not think it is inadequate as an autobiographical account, but it is not how one characterizes what is ultimately a false conclusion that was previously held true. One need not commit oneself to a particular choice either in the case of "victory" or "defeat", which are not themselves choices to be made.
Puzzle 2
T is true; therefore, evidence that it is false is false. This constitutes invalid reasoning, because it rules out new knowledge that may in fact render it truly false. It is also false to suppose that a human being is always capable of reasoning correctly under all states of knowledge, or even that they possess sufficient knowledge of a particular body of information perfectly so as to reason validly.
In general, one should not be concerned with how "misleading" a given amount of evidence is. To reason on those grounds, one could suppose a given bit of evidence would always be "misleading" because one "knows" that the contrary of what that bit of evidence suggests is always true. (The fact that there are people out there who do in fact "reason" this way, based on evidence, as in the superabundant source of historical examples in which they continue to believe in a false conclusion, because they "know" the evidence that it is false is false or "misleading", does not at all validate this mode of reasoning, but rather shores up certain psychological proclivities that suggest how fallacious their reasoning may be; however, this would not itself show that the course of necessary reasoning is incorrect, only that those who attempt to exercise it do so very poorly.) In the case that the one is dealing with a theorem, it must be true, provided that the reasoning is in fact valid, for theorematic reasoning is based on any axioms of one's choice (even though it is not corollarial). !! However, if the apodosis concerns a statement of evidence, there is room for falsehood, even if the reasoning is valid, because the premisses themselves are not guaranteed to be always true.
The proper attitude is to understand that the reasoning prior to exposure of evidence/reasoning from another subject (or one's own inquiry) may in fact be wrong, however necessary the reasoning itself may seemingly appear. No amount of evidence is sufficient evidence for its absolute truth, no matter how valid the reasoning is. Note that evidence here is indeed characteristic of observational criteria, but the reasoning based thereon is not properly deductive, even if the reasoning is essentially necessary in character. Note that deductive logic is concerned with the reasoning to true conclusions under the assumption that the relevant premisses are true; if one is taking into account the possibility of premisses which may not always be true, then such reasoning is probabilistic (and necessary) reasoning.
!! This, in effect, resolves puzzle 1. Namely, if the theorem is derived based on valid necessary reasoning, then it is true. If it isn't valid reasoning, then it is false. If "defeat" consists in being shown that one's initial stance was incorrect, then yes, it is essential that one takes the stance of having been defeated. Note that puzzle 2 is solved in fundamentally the same manner, despite the distracting statements ME, AME, and MF, on account of the nature of theorems. Probabilities nowhere come into account, and the employment of Bayesian reasoning is an unnecessary complication. If one does not take the stance of having been defeated, then there is no hope for that person to be convinced of anything of a logical (necessary) character.
"T is true; therefore, evidence that it is false is false. This constitutes invalid reasoning, because it rules out new knowledge that may in fact render it truly false."
Actually, I think if "I know T is true" means you assign probability 1 to T being true, and if you ever were justified in doing that, then you are justified in assigning probability 1 that the evidence is misleading and not even worth to take into account. The problem is, for all we know, one is never justified in assigning probability 1 to any belief. So I'd say the problem is a wrong question.
Edited: I meant probability 1 of misleading evidence, not 0.
I present here two puzzles of rationality you LessWrongers may think is worth to deal with. Maybe the first one looks more amenable to a simple solution, while the second one has called attention of a number of contemporary epistemologists (Cargile, Feldman, Harman), and does not look that simple when it comes to a solution. So, let's go to the puzzles!
Puzzle 1
At t1 I justifiably believe theorem T is true, on the basis of a complex argument I just validly reasoned from the also justified premises P1, P2 and P3.
So, in t1 I reason from premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
At t2, Ms. Math, a well known authority on the subject matter of which my reasoning and my theorem are just a part, tells me I’m wrong. She tells me the theorem is just false, and convince me of that on the basis of a valid reasoning with at least one false premise, the falsity of that premise being unknown to us.
So, in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
(R2) F, P1, P2 and P3
To the justified conclusion:
(~T) T is not true
It could be said by some epistemologists that (~T) defeat my previous belief (T). Is it rational for me to do this way? Am I taking the correct direction of defeat? Wouldn’t it also be rational if (~T) were defeated by (T)? Why ~(T) defeats (T), and not vice-versa? It is just because ~(T)’s justification obtained in a later time?
Puzzle 2
At t1 I know theorem T is true, on the basis of a complex argument I just validly reasoned, with known premises P1, P2 and P3. So, in t1 I reason from known premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
Besides, I also reason from known premises:
(ME) If there is any evidence against something that is true, then it is misleading evidence (evidence for something that is false)
(T) T is true
To the conclusion (anti-misleading evidence):
(AME) If there is any evidence against (T), then it is misleading evidence
At t2 the same Ms. Math tells me the same thing. So in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
But then I reason from::
(F*) F, RM and TM are evidence against (T), and
(AME) If there is any evidence against (T), then it is misleading evidence
To the conclusion:
(MF) F, RM and TM is misleading evidence
And then I continue to know T and I lose no knowledge, because I know/justifiably believe that the counter-evidence I just met is misleading. Is it rational for me to act this way?
I know (T) and I know (AME) in t1 on the basis of valid reasoning. Then, I am exposed to misleading evidences (Reliable Math), (Testimony of Math) and (F). The evidentialist scheme (and maybe still other schemes) support the thesis that (RM), (TM) and (F) DEFEATS my justification for (T) instead. So that whatever I inferred from (T) is no longer known. However, given my previous knowledge of (T) and (AME), I could know that (MF): F is misleading evidence. It can still be said that (RM), (TM) and (F) DEFEAT my justification for (T), given that (MF) DEFEAT my justification for (RM), (TM) and (F)?